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Some Concrete Aspects Of Hilbert's 17th Problem
 In Contemporary Mathematics
, 1996
"... This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction ..."
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Cited by 94 (4 self)
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This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction
Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity
 SIAM Journal on Optimization
, 2006
"... Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of ..."
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Cited by 76 (24 self)
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Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations. Key words.
Sums of squares of regular functions on real algebraic varieties
 Tran. Amer. Math. Soc
, 1999
"... Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a ..."
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Cited by 44 (9 self)
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Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth nonrational curve all of whose points at infinity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at infinity. The “compact ” case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at infinity, then every psd function is sos, provided the field R is archimedean. If R is not archimedean, there are counterexamples of genus 1.
There are significantly more nonnegative polynomials than sums of squares, arXiv preprint math.AG/0309130
, 2003
"... We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact ba ..."
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Cited by 30 (5 self)
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We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for the volumes of the bases. If the degree is greater than 2 then we show that the ratio of the volumes of the bases, raised to the power reciprocal to the ambient dimension, tends to 0 as the number of variables tends to infinity. 1
Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Cited by 19 (2 self)
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
Symmetric Positive 4 th Order Tensors & their Estimation from Diffusion Weighted MRI
, 2007
"... Abstract. In Diffusion Weighted Magnetic Resonance Image (DWMRI) processing a 2 nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DWMRI data. It is now well known that this 2 ndorder approximation fails to approximate complex local tissue ..."
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Cited by 16 (1 self)
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Abstract. In Diffusion Weighted Magnetic Resonance Image (DWMRI) processing a 2 nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DWMRI data. It is now well known that this 2 ndorder approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4 th order symmetric positive semidefinite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DWMRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semidefinite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4 th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert’s theorem on ternary quartics. This parametric representation is then used in a nonlinearleast squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higherorder tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DWMRI from an isolated rat hippocampus. 1
Lifting smooth curves over invariants for representations of compact Lie groups
 TRANSFORMATION GROUPS
, 2000
"... We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption. ..."
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Cited by 14 (10 self)
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We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization
 , in vol. Structured Matrices and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu
"... We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second par ..."
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Cited by 13 (3 self)
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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free ∗algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.
Frontiers of reality in Schubert calculus
 Bulletin of the AMS
"... Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fu ..."
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Cited by 10 (5 self)
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Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.